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SECTION I.-Of Planes, and of Bodies terminated by Planes.
195. IN the preceding sections, we have considered all the lines of any one of the figures which have been discussed, as in a plane surface ; that is, in a surface to which a straight line may be applied in every direction, so as to lie entirely in this surface. This has enabled us to unfold the elementary principles of linear geometry, and to measure the areas of surfaces bounded by straight lines or circular curves. We proceed now to investigate the geometrical relations in bodies. As the surfaces of many bodies consist, in whole or in part, of planes; and as all bodies involve the three dimensions of space, length, breadth and thickness; in discussing their geometrical character, it is necessary to consider not only the forms and magnitudes of planes, but their relative positions, their inclinations, and their relation to lines and points without them.* In discussing the general relations of lines to planes, and of planes to each other, we consider them as indefinitely extended, except where some limit is expressly stated. 196. If two straight lines are in the same plane, a second plane may be conceived to pass through one of these lines without coinciding with the other line ; it will therefore cut the first plane. But if any two points
* In what follows, the figures though represented on a plane, are considered as embracing the three dimenision of space. And where any line is supposed to pass behind any part of a body or of a plane, it is indicated in the figure by a dotted line.
common to the two planes, be taken, a straight line drawn between them will lie wholly in each of the planes (8); therefore—The interscetion of two planes is a straight line. 197. If the second plane be turned about the straight line through which it passes, till it coincides with the other line, it must also coincide with the other plane : Two straight lines, therefore, are sufficient to determine the position of a plane : And three points, not in the same straight line, will also determine the position of a plane; for the plane may be conceived to revolve about a straight line passing through two of the points; and if it be placed so as to coincide with the other point, it can revolve no farther without leaving it; it is therefore fixed. It follows from this that any two straight lines which cut each other, are in the same plane; for a plane may be passed through one of them, as AB (fig. 108), and Fig.108. turned till it coincide with the other CD. It is evident, therefore, that—any three points, not in the same straight line, being joined, two and two, by three straight lines, a plane triangle will be formed ; if four points are so joined, the quadrilateral, thus formed, may be a plane figure ; but if the plane passing through three of the points, do not at the same time pass through the fourth, the quadrilateral cannot be considered as a single surface. 19s. A straight line is said to be parallel to a plane, when it does not incline towards the plane, in either direction. A line parallel to a plane will therefore be, in every part, at the same distance from the plane; and consequently can never meet it, however far produced. A straight line is perpendicular to a plane when its inclination to the plane is the same on all sides. It is evident that the straight line will, in this case, be perpendicular to every straight line drawn in this plane, through the foot of the perpendicular. 199. , Let us suppose a straight line PD (fig. 109) Fig.109. to revolve about the straight line AB, always perpendicular to it, AB being considered as fixed. It is evident that PD, by this revolution, will generate a plane surface; for in whatever position the revolving line be placed, in the position PE, for instance, a plane may be conceived to pass through EPA, which being extended 7
will also pass through the opposite position of the revolv-
ing line, as PC. But as this line must be always perpen-
dicular to AP, the sum of the two angles APE, APC,
will be equal to two right-angles, and CPF will always
be a straight line, that is, the surface will be a plane, to
which AB will be perpendicular (8).
It follows, therefore, that—If two straight lines are
perpendicular to the same straight line, at the same point,
they are in the same plane perpendicular to this line.
200. Remark. It is evident that—There can be only
one line passing through P, perpendicular to this plane;
for if there could be another it must diverge from PA,
and therefore its inclination to the plane must be greater
on one side than the other, which cannot be the case
with a perpendicular (198).
201. It is also evident that—Through any point,
without a plane, as A for instance above the plane MN,
only one perpendicular can be drawn to that plane. For
a straight line passing through A in the same direction
with AP must be the same line ; another line, therefore,
which passes through A must be inclined to AP, and
consequently will be differently inclined to the plane
MN ; it cannot, therefore, be perpendicular to this
It is equally evident that—Only one plane can cut a
straight line at the same point, perpendicular to it. For
two planes passing through the same point, must be in-
clined to each other; they cannot, therefore, be both
perpendicular to the same straight line. *
202. As any two of the lines, CE, FD, passing through
the point P in this plane, are sufficient to fix the position
of this plane with respect to the line AB perpendicular
to these two lines (197); we say—If a straight line is
perpendicular to each of two straight lines drawn in a
plane through its foot, it is perpendicular to every other
straight line drawn through its foot in this plane, and is
therefore perpendicular to the plane.
203. Suppose AP to be perpendicular to the plane
MN (fig. 110); draw in this plane, from the foot of this
perpendicular, the equal straight lines, PC, PD ; and
draw AC, AD. The two triangles APC, APD, are both
right-angled at P; the sides containing the equal angles
are equal respectively; the triangles are therefore equal
by the second case, and give AC = AD. We therefore
say—Oblique lines drawn to a plane from any point
without it and equally distant from a perpendicular to
the plane, drawn through the same point, are equal.
204. It follows from the last result, that—If from any
point without a plane, a straight line be drawn perpendicular
to that plane, and also several equal oblique lines be drawn
to the plane from this point; these oblique lines will meet
the plane in the circumference of a circle whose centre is
the foot of the perpendicular. Consequently—If a
straight line be drawn perpendicular to a circle through
its centre, any point in this straight line is equally dis-
tant from every point in the circumference of the circle.
205. If we produce PD to G, and draw AG, it will be
easy to show that AG must be greater than AD (51),
and thence to show that—Of two oblique lines, falling
upon a plane at unequal distances from a perpehdicular,
that is the greater which falls at the greater distance
from the perpendicular.
206. Suppose the line AG (fig. 11 l) to be oblique to
the plane MN; draw from A to the plane the perpen-
dicular AP; join PG, and through G draw BC perpen-
dicular to PG. Then taking GC equal to GB, draw PB,
PC, AB, AC ; PB and PC will be equal (41), and AB
and AC will therefore be equal; and AG will be a straight
line drawn from the vertex of an isosceles triangle to the
middle of the base; that is—BC is perpendicular to the
oblique line AG, when it is perpendicular to the straighf
line which joins the foot of the oblique line with the foot
of the perpendicular A.P.
In this case AP and BC are said to be perpendicular
to each other, though they cannot meet.
207. Suppose the plane OP (fig. 112) to cut the plane
MN, in the line AB; draw through the point C, CD in the
plane MN, and CE in the plane OP, each perpendicular
to the intersection AB of these planes; then turning the
plane OP about the intersection till the two planes coin-
cide, CE will coincide with CD. If the part E of the plane
OP be now raised from the plane MN, turning this plane
about the intersection, this point will describe the arc
DE, which in every stage of the process will measure
the inclination of the two planes; the centre of this arc
is C; therefore–To measure the angle which two planes
make with each other, draw in each plane, through the
same point, a straight line perpendicular to the intersec-
tion, and the angle which these two lines make with each
other is the angle of the planes.
208. Remark. The angle made by two planes is call-
ed a diedral angle, that is, an angle of two faces; and is
designated by four letters, of which the two middle ones
are in the intersection, and the two others are in the
different planes out of the intersection ; as the diedral
angle PABN. The intersection AB of the two planes is
called the edge of the diedral angle. When this angle
is 900, the inclination of the plane PO to the plane MN,
is the same on each side of the intersection, and the
planes are said to be perpendicular to each other.
It is evident that two diedral angles are to each other
as the arcs of the plane angles which are their measures.
It is also manifest that two diedral angles which are op-
posite at the edge, as MABO and PABN, are equal.
209. Suppose the two planes CD and EF (fig. 113)
to be each perpendicular to the plane AB; from the
point G where the three planes meet, draw EH perpen-
dicular to the plane AB ; it will be in each of the planes
CD and EF (207); therefore—The intersection of two
planes each perpendicular to a third plane, is also per-
pendicular to this third plane.
210. If two straight lines are perpendicular to the
same plane, they have the same direction in space and
are therefore parallel to each other. Let CE be perpen-
dicular to the plane MN (fig. 114), and BD be par-
allel to CE; let the plane AD pass through the par-
allels; this plane will be perpendicular to the plane
MN (207), and DB parallel to EC, will be perpendicular
to the intersection AB; it will therefore have the same
inclination to the plane MN, as the plane AD has to the
plane MN; and as this inclination is a right angle, DB
is perpendicular to the plane MN. Therefore—A
straight line parallel to a second which is perpendicular
to any plane, is also perpendicular to this plane.
211. Take the two planes AB and CD (fig. 115) per-
pendicular to the same straight liue GH; draw GR and
GL in the plane AB, and from the point H draw HM
and HS parallel respectively to GL and GR : G.I. and
GR will be both perpendicular to the line GH, and con-
sequently will lie in the plane AB (201). Now the two
lines HM and HS determine the position of the plane
CD (198), and GL and GR determine the position of the